TGr 


UC-NRLL 


3ft? 


IN  MEMORIAM 
FLOR1AN  CAJOR1 


THIIE 

MECHANICAL  ERRORS 


-IN- 


Tfie  Common  Ttieonj  of  Flexure 


5 
c- 


THE 


MECHANICAL  ERRORS 


-IN- 


Tfie  Common  THBoru  of  Flexure 


•BY- 


E.    H.  [COUSINS, 

C1VII,  ENGINEER. 


AUSTIN,  TEXAS: 

EUGENE  VON  BOECKMANN,  PRINTER  AND  BOOKBINDER. 
'    1891. 


CAJORI 


General    Principles    Applicable    to 
Balanced  Parallel  Forces. 


Before  discussing  the  mechanics  involved  in  the  "com- 
mon theory  of  flexure,"  it  is  necessary  to  restate  some  ele- 
mentary principles  relating  to  the  action  of  forces,  and  to 
define  certain  distinctions  in  technical  terms,  drawn  from 
practical  mechanics,  that  are  not  found  in  text-books  of  the 
subject. 

i  st.  In  a  balanced  system  of  parallel  forces  the  sums  of 
the  forces  acting  in  opposite  directions  must  be  zero ;  in 
other  words,  the  sum  of  the  vertical  forces  must  be  zero  and 
the  sum  of  the  horizontal  forces  must  be  zero. 

2nd.  The  algebraic  sum  of  the  moments  relative  to  any 
axis  shall  be  nothing. 

3d.  When  two  equal  and  opposite  forces  balance,  whose 
lines  of  action  do  not  coincide,  a  shearing  force  must  be 
transmitted  from  the  line  of  action  of  one  to  that  of  the 
other,  this  shearing  is  the  right  line  balancing  agent. 

4th.  The  lever-arm  of  a  force  is  the  distance  traveled  by 
its  shearing  force  in  opder  to  reach  a  centre  of  motion  or 
tendency. 

5th.  A  shearing  force  cannot  be  transmitted  by  a  body 
through  and  beyond  a  center  of  rotation  or  tendency. 

6th.  A  shearing  force  has  no  leverage  and  therefore  no 
moment. 

7th.  A  force  to  possess  leverage  must  transmit  its  shear- 
ing force  to  a  center  of  rotation  or  tendency,  where  it  must 
be  opposed  by  an  equal  and  opposite  force. 

8th.     A  body,   and   therefore  any  part  of  the  body,   is  at 


rest  in  any  direction,  when  each  particle  of  one  of  its  planes 
is  solicited  to  move  in  opposite  directions  by  pairs  of  equal 
and  opposed  conspiring  forces,  parallel  to  the  given  direction. 

9th.  Two  equal  and  opposed  shearing  forces  will  pro- 
duce compression  in,  and  if  sufficiently  intense,  crush  the 
body  that  transmits  them. 

loth.  When  an  unbalanced  force  is  applied  to  a  beam 
and  equilibrium  is  established  by  the  development  of  inter- 
nal forces,  the  originally  unbalanced  force  must  be  equili- 
brated by  its  own  generated  components,  and  not  by  those 
of  some  other  applied  force. 

The  ist  and  2nd  conditions  are  those  usually  given  as 
solely  sufficient  to  determine  the  conditions  of  equilibrium  of 
any  body,  the  others  are  none  the  less  universally  true,  and 
are  here  formulated  for  the  convenience  of  reference,  as  their 
existence  is  very  frequently  ignored  by  writers,  in  their  dis- 
cussion of  mechanical  problems. 

Moments. — The  moment  of  a  force  about  a  point  meas- 
ures the  tendency  of  the  force  to  produce  rotation  about  the 
point,  when  one  point  of  the  body  is  "fixed"  the  tendency 
to  rotation  is  around  this  point,  which  is  therefore  the  origin 
of  moments. 

Theoretical  mechanics  takes  the  moment  of  a  force  with 
respect  to  any  point,  whether  there  is  a  tendency  to  rotate 
around  it  or  not,  in  the  first  case  the  moments  possess  lever- 
age or  mechanical  power,  in  the  second  they  do  not;  theo- 
retically, the  moment  of  a  force  may  have  an  infinite  num- 
ber of  values,  but  in  practical  mechanics  it  has  value  only 
when  taken  with  reference  to  a  center  of  rotation  or  tendency 
in  the  body  to  which  it  is  applied. 

Theoretical  moments,  from  an  "anywhere  origin,"  are 
sufficient  to  determine  the  conditions  under  which  a  whole 
body  may  be  at  rest,  relatively  to  the  other  bodies  of  the 
world;  but  when  we  equate  power  and  resistance,  as  in  a 
solid  beam,  the  moments  must  possess  mechanical  power, 
such  as  Archimides  would  have  used  had  he  turned  the 
world  over,  which  power  they  only  have  when  taken  with 


reference  to  a  common  center \  around  which  they  all  tend  to 
rotate,  and  around  which  the  body  could  rotate  if  the  forces 
that  established  the  condition  of  tendency  were  removed. 
The  location  of  this  center  clearly  classes  our  forces  into 
poivet  and  resistance,  the  failure  to  do  this  in  the  "common 
theory"  causes  the  tension  to  be  the  resistance,  and  \hepower 
to  be  the  load  and  the  compression,  as  their  moments  add, 
having  like  signs,  which  of  itself  condemns  the  theory. 

In  nature,  the  moments  of  two  applied  forces,  whose 
signs  are  unlike,  never  add,  as  the  mechanical  power  ex- 
pressed by  each  moment  tends  to  do  opposite  or  unlike 
things.  The  only  exception  to  this  invariable  rule  is  on 
paper,  where  the  moments  of  the  unbalanced  forces  of  a 
static  couple  are  said  to  add  ;  but  the  moments  of  these 
forces  possess  no  mechanical  power,  only  a  possibility  of 
such,  which  is  only  realized  when  the  forces  balance  as 
right  line  forces,  and  then  one  force  has  no  moment  as  its 
lever-arm  is  zero.  The  application  of  this  mythical  mechan- 
ical moment  of  the  unbalanced  forces  of  a  couple  to  the 
tension  and  compression  in  a  solid  beam  has  lead  to  a  palpa- 
ble violation  of  the  fundamental  principle  above  stated. 
The  moment  given  a  static  couple  violates  the  condition  of 
tendency  which  is  the  very  essence  of  moments. 

Shearing  Force  and  Leverage. — Shearing  is  a  peculiar 
auxiliary  force  that  is  found  to  exist  when  an  applied  force 
has  leverage  ;  it  travels  from  the  point  of  application  of  the 
force  to  the  center  of  rotation  or  tendency  to  such  rotation, 
it  enables  the  force  to  possess  leverage  and  to  balance  an- 
other opposed  equal  force  when  their  lines  of  action  do  not 
coincide. 

At  any  section  between  the  point  of  application  and  the 
center  of  rotation,  it  is  equal  in  intensity,  parallel  to  the 
force,  and  has  the  same  direction — its  tendency  is  to  cross- 
cut the  body  that  transmits  it  at  each  sectional  plane. 

A  shearing  force  cannot  possess  leverage,  its  existence 
being  the  result  of  leverage,  it  cannot  in  turn  reproduce  its 
own  cause.  ' 


The  forces  that  have  a  shearing  force  are  purely  static, 
such  as  a  single  force  that  tends  to  produce  rotation,  and  a 
pair  of  opposed  tensile  forces  that  tend  to  resist  rotation, 
if  these  forces  have  no  shearing  they  neither  tend  to  produce 
nor  to  resist  rotation  ;  for  they  then  have  no  leverage. 
But  neither  of  the  two  opposed  compressive  forces  have  any 
shearing  force,  and  the  leverage  they  possess  is  direct  with- 
out its  assistance.  A  compressive  force  has  this  leverage 
in  resisting  the  rotating  tendency  of  another  force,  only 
when  their  tendencies  are  in  opposite  directions  around  an 
actual  (not  theoretical)  center  of  tendency  to  rotation.  In 
Fig.  i,  if  the  actual  center  of  rotation  is  below  the  point 
of  application  of  the  compressive  force  C,  as  it  really  is, 
then  the  force  C  has  no  leverage  in  resisting  the  rotating 
tendency  of  the  load  I,. 

Observation  teaches  that  a  body  cannot  transmit  a  shear- 
ing force  a  greater  distance,  than  that,  from  the  line  of  action 
of  the  force  to  the  center  of  rotation,  and  that  this  distance 
traveled  by  the  shearing  is  the  limit  to  the  length  of  the 
lever-arm  of  the  force. 

The  beam  A,  Fig.  i  presses  the  support  S  through  the 
shearing  force  I,  and  the  support  presses  the  beam  with  an 
equal  and  opposite  force,  as  this  force  S  may  also  be  the 
shear  of  another  force,  these  two  equal  and  opposed  shears 
will  produce  pressure  equal  to  that  had  the  two  forces 
been"  directly  applied  at  S.  A  familiar  illustration  of  this 
is  seen  in  the  ordinary  nut-cracker,  the  pressure  on  the  nut 
is  the  sum  of  the  shears  of  the  force  applied  by  the  hand 
and  the  force  developed  at  the  rivet  of  the  hinged  joint. 

Couples. — Two  equal  and  opposite  unbalenced  facts,  whose 
lines  of  action  do  not  coincide  constitute  a  static  couple, 
and  the  bod)''  to  which  it  is  applied  should  be  in  equili- 
brium, as  the  ist  condition  is  rigidly  fufilled,  but  the  ef- 
fect of  a  couple  is  to  rotate  the  body  about  its  center  of 
mass,  hence  it  is  not  in  equilibrium.  However,  the  in- 
stant the  body,  from  any  cause,  is  enabled  to  transmit  a 
shearing  force,  or  fulfill  our  3d  condition,  equilibrium  is  es- 


tablished,  but  the  pair  of  forces  cease  to  form  a  couple  and 
become  a  part  of  a  common  lever  where  one  force  is  the 
power,  the  other  the  fulcrum  and  the  resistance  whatever 
enabled  the  body  to  transmit  a  shearing  force. 

A  "static  couple"  is  a  misnomer,  the  rotation  produced 
by  it  about  the  "center  of  mass"  is  dynamic,  which  cannot 
be  converted  into  a  tendency  about  the  same  "center  of 
mass' '  for  the  instant  it  becomes  static  equilibrium  the  cen- 
ter of  tendency  to  rotation  shifts  from  the  center  of  mass  to 
the  point  of  application  of  one  of  the  forces  that  previously 
formed  the  "static  couple."  The  application  of  the  "parel- 
lelogram  of  forces"  to  the  equilibrated  forces  will  demon- 
strate the  truth  of  this,  or  it  fulfills  our  yth  condition. 

Theoretical  mechanics  considers  any  two  equal  and  oppo- 
site forces '  to  form  a  couple  that  either  rotates  or  tends  to 
rotate  the  body  about  its  center  of  mass,  whether  the  forces 
are  balanced  as  right  line  forces  or  not, '  This  is  evidently 
erronious,  for  if  the  forces  are  balanced  it  is  physically  im- 
possible for  the  body  to  transmit  the  shearing  force  (the 
right  line  balancing  agent)  through  a  center  of  rotation  or 
tendency  to  such  rotation,  it  can  transmit  it  to  such  a  cen- 
ter, but  not  through  it,  hence  if  the  forces  tend  to  turn  the 
body  about  its  center  of  mass  they  are  unbalanced.  This 
is  an  especially  important  distinction  to  make,  since  a 
writer,  who  is  usually  very  clear,  states  that  the  forces  (I, 
S)  Fig.  i  form  a  couple  that  tends  to  rotate  the  body  A, 
about  its  center  of  mass,  thus  making  the  shearing  force  L 
travel  through  a  center  of  tendency  to  rotation,  and  a  point 
that  is  "fixed"  by  the  opposite  forces,  S,  and  shearing  force 
I,,  tend  to  travel  in  the  arc  af  a  circle  around  this  center, 
both  of  which  being  a  mechanical  impossibility.  The  rule 
of  statics  so  often  quoted  by  authors  in  the  analysis  of  the 
common  theory,  that  "if  two  couples  applied  to  the  same 
body  balance  each  other  they  must  have  opposite  algebraic 
signs,"  and  its  corallary,  if  the  forces  in  the  direction  of  one 
axis  reduce  to  a  couple  those  in  the  direction  of  the  other 
must  reduce  to  a  couple  also,  for  equilibrium  has  led  them 


to  erronious  conclusions.  The  rule  is  only  true  when  the 
pairs  of  forces  balance  as  right  line  forces,  and  thus  cease 
to  be  static  couples,  and  the  resultant  moment  of  each  pair 
is  the  leverage  moment  of  a  single  force  and  its  corallary, 
when  the  resultant  moment  of  one  pair  reduces  to  the  mo- 
ment of  a  single  force,  the  resultant  moment  of  the  other 
pair  must  reduce  to  the  leverage  moment  of  a  single  force 
with  opposite  sign. 

To  illustrate  (L  S)  and  (T  C)  Fig.  i  are  said  to  form 
such  couples.  Opposed  to  S  there  is  a  shearing  force  Iy 
equal  to  it,  our  couple  (I,  S)  in  balancing  the  forces  has 
the  vertical  effect  of  three  forces,  the  resultant  moment  of 
which,  according  to  a  rule  of  statics,  is  the  moment  of  the 
single  force  L,.  In  the  same  way,  horizonally,  we  have  the 
effect  of  three  forces,  T  and  C  and  shearing  force  T,  equal 
and  opposed  to  C,  the  resultant  moment  of  the  three  is  the 
moment  of  the  single  force  T. 

When  the  resultant  moment  in  the  direction  of  one  axis 
does  not  reduce  to  that  of  a  single  force,  as  the  tension  and 
compression  in  a  beam,  the  mechanical  necesssity  for  and 
their  ability  to  develop  equally  does  not  exist.  In  this  case 
the  line  of  direction  of  neither  force  passes  throngh  the  cen- 
ten  of  rotation.  The  line  of  action  of  the  compressive  force  C 
being  between  the  line  of  action  of  its  companion  force  T 
and  the  center  of  rotation,  the  point  of  support,  its  moment 
and  that  of  the  rotating  force  must  have  opposite  signs,  and 
T  must  have  the  same  sign  and  direction  of  C,  if  it  is  an 
"applied"  force. 

THE  COMMON  THEORY  OF  FLEXURE. 

It  is  a  misnomer  to  call  the  common  theory  '  'a  theory  of 
flexure' '  for  the  theory  itself  does  not  recognize  the  effect 
that  is  produced  upon  the  interior  of  the  beam  by  bending 
in  the  slightest  degree.  It  is  true  from  knowledge  obtain- 
ed outside  of  the  teaching  of  the  theory,  the  total  amount 
of  the  interior  forces  is  derived  from  considering  them  to 
vary  uniformly  from  a  neutral  line,  but  this  is  also  done 


when  the  beam  is  not  bent  at  all,  then  it  can  no  more  be 
called  a  '  'a  theory  of  flexure' '  than  it  is  of  any  other  struc- 
ture in  which  the  stresses  are  equal,  such  as  bridge  trusses, 
etc. 

To  be  a  theory  of  flexure  it  must  recognize  that  the  stres- 
see  before  flexure  begins,  exist  in  a  different  condition  from 
that  after,  and  that  flexure  caused  them  to  change  this  con- 
dition and  to  assume  the  order  of  arrangement  that  is 
known  to  exist  in  a  bent  beam. 

In  its  mechanics  the  common  theory  is  based  up  undis- 
puted principles  of  pure  statics,  but  when  it  is  attempted  to 
explain  the  phenominon  known  to  attend  flexure  it  can 
only  be  done  by  adopting,  at  every  point,  principles  just 
the  reverse  of  its  teachings,  and  it  is  a  mechanical  error  to 
attribute  to  it  effects  that  it  cannot  produce. 

In  reasoning  from  an  effect  to  its  cause  the  human  mind 
more  readily  accepts  false  logic  as^true  than  when  it  adopts 
the  reverse  process.  The  best  authorities  emphasize  the 
statement  "that  a  body  in  static  equilibrium  is  conceived  to 
be  incaple  of  bending,  breaking  or  changing  its  shape  in  any 
way,  and  their  effect  is  not  considered,"  yet  from  the 
knowledge  of  what  takes  place  in  "bending,"  "those 
who  ought  to  know  how  to  apply  principles"  to  effect  has 
described  its  cause  as  being  '  'a  clear  case  of  statics' '  when 
the  reverse  process  of  reasoning  would  have  denied  the  ef- 
fect claimed  at  every  point.  To  the  extent  that  a 
beam  rotates  or  bends  under  a  transverse  or  a  column  de- 
flects under  a  longitudinal  load,  it  is  actuat  constrained  rota- 
tion, this  induces  actual  longitudinal  molecular  movement, 
which,  of  itself,  takes  it  out  of  the  jurisdiction  of  static 
laws.  Static  equilibrium  is  only  a  tendency  to  rotation, 
which  rotation,  if  it  took  place,  would  be  free  without  mo- 
lecular movement. 

Static  forces  acting  in  a  single  plane  are  always  equili- 
brated on  the  principle  of  the  parallelogram  of  forces,  when 
their  resultants  intersect.  In  the  parallelogram  only  two 
forces  possess  mechanical  power,  and  the  equality  of  their 


10 

moments  prevents  rotation,  but  the  common  theory  con- 
ceives that  three  forces  have  this  power  ;  as  the  compres- 
sion force  simply  prevents  horizontal  translation,  it  may  be 
left  out,  just  as  the  companion  force  that  prevents  vertical 
translation,  as  neither  have  any  lever-arm. 

In  the  "parallelogram  of  forces"  the  center  of  tendency 
to  rotation  is  at  the  end  of  the  diagonal  where  the  forces  are 
balanced  by  their  reactions,  which  in  the  solid  beam  is  a 
point  on  the  resultant  of  the  compressive  force,  but  there 
can  be  no  rotation  around  this  point  and  therefore  no  ten- 
dency as  the  two  are  inseparable,  which  destroys  the  com- 
mom  theory  equilibrium,  as  it  must  be  around  some  point 
outside  of  the  parellelogram. 

Generated  forces  that  produce  compressive  strain  and 
possess  mechanical  power  or  leverage  are  not  recognized  in 
statics,  and  its  treaties  make  no  attempt  to  trace  the  me- 
chanical relation  that  must  exist  between  the  generating 
and  generated  forces,  therefore  the  rules  that  formulate 
the  principles  of  statics  do  not  apply  when  this  is  the  case, 
or  more  than  two  intersecting  forces  possess  mechanical 
power  in  an  equilibrated  system. 

The  "common  theory  of  flexure"  is  deduced  from  the 
following  conditions.  L,et  a  beam  be  loaded  at  its  ends 
and  supported  at  the  center,  then  after  equilibrium  has 
been  established  conceive  one-half  of  the  beam  to  be  re- 
moved, and  the  balanced  condition  destroyed.  Then  it  is 
conceived  that  we  have,  only,  an  unbalanced  force  L,  and 
a  rigid  body  A,  and  that  we  must  by  external  applied 
forces  re-establish,  de  novo,  the  equilibrium  that  existed  be- 
tween the  external  force  I,  and  the  developed  internal 
forces,  which  we  destroyed  in  the  cutting  process. 

This  gives  the  diagram  of  forces  represented  in  Fig.  i. 
From  the  ist  and  2nd  conditions  given,  we  have  L,=S,  and 
T=C,  and  the  moment  of  the  (L,=S),  system  equal  to  the 
moment  of  the  (T=C)  system. 

That  T  must  equal  C  as  the  sole  and  sufficient  cause  of 
equilibrium  is  only  true  when  the  forces  are  concurrent  and 


11 


their  lines  of  action  coincide,  but  here  a  distance  intervenes 
between  them,  and  in  order  to  balance  they  must  get  to- 
gether in  some  manner,  and  be  opposed  just  as  if  they 
were  concurring  and  their  lines  of  action  coincided, — this 
the  beam  A  enables  them  to  do  by  transmitting  a  longitudi- 
nal shearing  force  from  T  to  C. 

To  follow  the  line  of  argument  of  a  writer1  on  this  sub- 
ject— conceive  that  we  have  only  the  forces  L  and  T,  verti- 
cally, the  ist  condition  must  be  fulfilled,  viz:  "L,=S  since 
the  pressure  upon  the  fulcrum  must  equal  the  dormant 
force  L,"  or  its  shearing  force.  "To  prevent  rotation  we 


A 


H- 


-  C 


\/ 


I 


Fig.  1 


must  now  have  1,1=  Td.  But  these  two  conditions  are  not 
sufficient  unless  we  introduce  a  new  force  C  from  outside 
and  make  C=T,  we  shall  have  horizontal  motion."  From 
his  failure  to  state  how  T  balanced  C,  the  writer  here  loses 
the  thread  of  his  argument  and  thus  reaches  a  conclusion 
entirely  at  variance  with  the  premises  laid  down  by  him- 
self. He  should  have  said,  to  be  consistent,  "the  pressure 
of  the  half  beam  A  against"  and  opposite  the  "new  force" 
C  must  be  equal  to  the  '  'dormant'  '  or  shearing  force  T. 
And  as  shearing  force  T,  that  is  equal  and  opposed  to  C,  is 
not  a  "new  force"  but  the  same  force  T  in  dnew  place,  he 
would  have  reached  the  conclusion  that  the  load  I,  has  only 
one  horizontal  component  instead  of  two,  and  the  moment 


(i)  Prof.  DuBois,  Bng.  News,  Jan.  2ist,  1888. 


12 

of  resistance  is  the  moment  of  a  single  stress  instead  of  that 
of  a  stress  couple." 

The  writer  also  erred  in  giving  the  ist  condition  as  his 
reason  for  making  T— C  to  prevent  horizontal  "traveling", 
in  the  whole  beam,  that  is  accomplished  from  the  forces 
complying  with  our  8th  condition,  which  is  solely  sufficient, 
and  it  was  a  fatal  error  to  "conceive"  that  the  part  could 
"travel"  independently  of  the  whole.  His  true  reason  for 
this  is  found  in  our  yth  condition,  in  order  that  T  may  pos- 
sess leverage  to  resist  rotation. 

The  advocates  of  the  "common  theory  of  flexure"  insist 
that  the  tension  must  balance  the  compression  in  the  beam, 
just  as  the  force  at  the  supports  balances  the  load,  and  as 
their  text-books  on  mechanics  demonstrates  no  other 
method  of  balancing  except  through  the  aid  of  an  auxiliary 
shearing  force,  then  they  must  accept  all  of  the  absurdities 
and  contradictions  of  this  method  when  applied  to  a  solid 
beam. 

Distribution  of  the  compression. — It  is  rightly  claimed  by 
"the -common  theory"  that  at  any  section  the  tension  is 
distributed  through  the  portion  of  the  beam  that  it  occu- 
pies as  an  uniformly  varying  force.  When  the  resultant  T 
balances  C  the  laws  of  mechanics  require  that  the  shearing 
force  T  shall  be  greatest  at  nt  Fig.  2,  where  the  common 
theory  requires  its  opposing  force  to  be  zero.  The  vertical 
shearing  force  is  a  maximum  at  the  inner  side  of  the  sup- 
ports, this  horizontal  shearing  force  must  follow  the  same 
law,  and  be  uniformly  distributed  along  the  line  ns  instead 
of  as  an  uniformly  varying  force.  Wiesbach  states  that  n 
is  the  center  of  rotation — this  limits  the  lever-arm  of  T  to 
its  distance  from  n,  and  this  will  be  all  of  the  moment  that 
sustains  the  load,  as  it  is  force  T  that  posseses  leverage 
and  not  shearing  or  '  'dormant' '  force  T. 

The  neutral  line  destroyed. — The  method  of  balancing  T 
and  C,  adopted  by  '  'the  common  theory' '  requires  that  two 
equal  and  opposite  shearing  forces  T  (or  C's  after  they  be- 
came opposed)  shall  pass  through  the  neutral  line,  the  ef- 


13 


feet  of  which  will  be  to  destroy  the  properties  claimed  for 
the  line  by  the  theory. 

Is  it  not  a  mechanical  absurdity  to  say  that  T  and  C 
balance  each  other  horizontally  as  right  line  forces,  when 
their  lines  of  action  do  not  coincide  and  that  their  is  be- 
tween their  lines  of  action  a  neutral  or  dead  line  over  which 
the  effect  of  neither  force  can  pass  ?  The  equal  but  unbal- 
anced forces  of  a  "static  couple"  that  simply  overcomes 


/\ 


the  inertia  of  the  body  and  causes  it  to  rotate  about  its  cen- 
ter of  mass  has  what  resembles  the  neutral  line,  but  it  dis- 
appears as  soon  as  the  forces  balance  each  other  by  the 
bodies  transmitting  a  shearing  force. 

A  solid  beam  has  a  neutral  line  and  the  compression 
varies  uniformly,  neither  of  which  could  exist,  as  such  in 
the  beam,  at  the  same  time  with  a  longitudinal  shearing 
force.  The  tension  cannot  balance  the  compression  with- 
out this  force,  therefore  they  do  not  balance,  and  the  theory 
must  be  reversed  in  order  to  explain  the  facts. 

Method  of  Sections,  or  of  Substituting  '  *  applied' '  for  gener- 
ated Forces.—  The  Railroad  Gazette2  says  of  my  objection 
to  the  establishment  of  the  "common  theory"  by  what  is 
known  as  the  "method  of  sections"  that  "this  mode  of  pro- 
cedure is  so  well  understood  in  the  treatment  of  stress  that 


(2)  September  6th,  1889. 


14 

it  seems  unnecessary  to  refer  to  it  unless  we  have  made  an 
error  in  its  application,"  this  error  I  think  is  evident. 

1  'The  applied  forces  and  horizontal  stresses  on  each  side 
of  any  section  of  a  loaded  beam  hold  each  other  in  equili- 
brium" each  on  its  own  side  of  this  section.  Now  if  we  con- 
ceive one  part  of  the  loaded  beam  to  be  removed,  at  any 
section,  and  the  existing  equilibrium  in  the  part  that  re- 
mains to  be  destroyed,  or  it  is  reduced  to  a  rigid  body  and 
an  unbalanced  load,  then  the  horizontal  forces  that  we 
must  "apply"  at  this  section  to  establish  horizontal  (right 
line)  and  rotary  equilibrium,  at  the  same  time  gives  us  '  'a 
clear  case  of  statics."  This  is  the  "well  recognized"  con- 
dition of  equilibrium  when  the  vertical  applied  force  gener- 
ates nothing  but  free  rotation  or  its  tendency,  and  it  cer- 
tainly cannot  be  identical  with  its  condition  when  it  gener- 
ates constrained  rotation  accompanied  by  horizontal 
stress  through  "deformation,"  therefore  the  "common  the- 
ory" leaves  out  of  consideration,  entirely,  the  effect  of  the 
generation  upon  the  generating  cause  when  it  substi- 
tutes-the  right  handed  applied  couple  (T  C),  Fig.  i  for  the 
left  handed  generated  couple  in  A,  for  as  mechanical 
powers  they  are  of  entirely  reverse  action. 

But  if  we  conceive,  as  we  must,  that  the  same  equili- 
brium that  existed  in  the  half  beam  before  removal  must 
exist  in  it,  after  removal,  as  we  have  taken  away  neither 
the  cause,  the  "applied"  load,  nor  the  effect,  the  generated 
stress,  on  its  own  side  of  the  section,  which  equilibrated 
each  other  in  the  whole  beam,  and  therefore  must  do  so  in 
the  half  beam,  then  it  is  not  such  "a  clear  case  of 
statics."  In  each  halt  of  the  beam,  after  being  divided,  we 
will  have  a  vertical  applied  force,  whose  moment  has  been 
exhausted  or  equilibrated  by  the  moment  of  its  two  gener- 
ated horizontal  components,  otherwise  our  cutting  and 
removing  will  introduce  an  element  that  did  not  exist  in 
the  loaded  beam  as  a  whole,  viz:  the  unbalanced  moment  of 
the  vertical  load,  which  we  are  not  warranted  in  doing  for 
the  pleasure  of  balancing  if  again  in  some  other  manner. 


15 

The  bending  of  a  beam,  under  a  tranverse  load,  is  actual 
rotation  around  some  point,  and  not  simply  a  tendency 
around  every  known  point,  were  it  not  so  the  stresses  could 
not  arrange  themselves  on  each  side  of  the  neutral  line  as 
as  they  are  known  to  do.  Statics  does  not  recognize  that 
there  can  be  any  motion  in  an  equilibrated  system  of  forces 
by  reason  of  the  "deformation"  of  the  body  upon  which  it 
acts,  but  conceives  that  the  body  cannot  bend  or  break  un- 
der the  action  of  any  force,  however,  intense,  but  always 
retains  its  size  and  shape  unchanged. 

If  the  rotation  of  the  beam  A,  Fig.  i  by  the  load  L,,  gen- 
erated nothing  but  free  rotation  or  tendency,  which  is  all 
that  is  contemplated  by  statics,  then  "the  common  theory" 
method  of  balancing  by  the  section  method  is  correct.  But 
the  rotation  or  tendency  instead  of  being  free,  where  every 
point  in  the  body  tends  to  move  in  the  arc  of  a  circle  with 
a  radius  equal  to  its  distance  from  the  common  center,  it  is 
constrained  rotation  where  every  point  actually  moves  in  a 
curved  line,  with  a  constantly  changing  distance  from  '  'the 
common  center, ' '  from  each  such  point  having  a  longitudi- 
nal motion. 

In  free  rotation  or  its  tendency  the  load  generates  no 
stress  components  as  there  is  no  horizontal  motion,  and  its 
tendency  to  freely  rotate  the  beam  in  one  direction  must  be 
resisted  by  outside  "applied"  forces  that  tend  to  rotate  it, 
equally,  in  the  opposite  direction,  as  in  the  common  theory, 
but  the  constrained  rotation  of  the  beam  generates  two  hor- 
izontal stress  components,  and  the  effort  expended  by  the 
load  in  their  generation  equilibrates  and  absorbs  its  rotat- 
ing power,  hence  there  is  no  unbalanced  rotating  tendency 
for  our  outside  "applied"  forces,  at  the  section,  to  balance 
by  rotating  the  beam  in  the  opposite  direction. 

The  method  of  sections  will  lead  to  no  error  if  it  is  sim- 
ply used  to  interpret  the  equilibrium  that  is  established 
through  the  generation  of  horizontal  forces  instead  of  es- 
tablishing it  de  novo  as  is  done  in  the  "common  theory." 
In  this  interpretation  two  separate  and  distinct  uses  must 


16 

be  made  of  the  "applied"  forces, — ist,  to  apply  such  forces 
that  will  establish  the  horizontal  equilibrium  that  existed 
in  the  whole  beam,  2nd,  to  apply  such  forces  that  their  me- 
chanical power,  or  moment,  will  resist  the  rotating  power 
of  the  load  in  the  same  identical  way  that  it  was  equilibrat- 
ed by  the  moments  of  the  generated  stresses  on  its  own  side 
of  the  section  of  division,  and  both  objects  cannot  be  ac- 
complished at  the  same  time  by  the  same  diagram  of  forces 
as  is  done  in  the  "common  theory." 

The  first  problem  then  is  to  apply  at  the  removed  section 
two  such  force  systems,  of  which  T  and  C  are  the  result- 
ant Fig.  i,  that  will  enable  the  half  beam  A  to  sustain  the 
load  L,  with  the  same  generated  stresses  in  A  that  existed 
before  the  removal  of  the  other  half  of  the  beam.  After 
division  the  horizontal  stresses  must  be  conceived  to  exist 
in  each  part,  just  as  they  did  before  division,  and  as  it 
would  be  impossible  for  these  stresses  to  import  motion  to 
the  half  beam  in  which  they  exist,  whether  they  are  equal 
or  not,  there  is  no  mechanical  necessity  for  our  making 
our  applied  forces  equal.  We  must  then  find  only  such 
forces  that  must  have  existed  in  the  removed  portion  of  the 
beam  in  order  that  such  forces  when  applied  will  produce 
with  the  forces  already  in  the  half  beam,  the  observed 
horizontal  deformation,  that  is,  the  elongation  by  tension 
and  the  shortening  by  compression  and  the  original  hori- 
zontal equilibrium  will  be  restored  by  applying  a  pull  equal 
and  opposite  to  the  existing  pull,  and  a  push  equal  and  op- 
posite to  the  existing  push,  and  this  is  the  only  condition 
we  can  legitimately  impose  upon  these  forces,  when  they 
are  equal  opposed  pairs  and  conspiring  at  any  section. 
"The  applied"  forces  T  and  C,  Fig.  i  represent  in  the  sys- 
tem the  removed  load  on  B,  and  thus  enables  A  to  sustain 
the  same  load  that  it  did  before  removal  and  with  the  same 
generated  stresses  in  A. 

The  second  problem  then  is  to  find  two  such  forces  that 
when  substituted  for  the  left  handed  couple  generated  in  A, 
by  the  load  I,,  Fig.  i,  the  mechanical  action  of  the  substi- 


17 

tuted  or  '  'applied"  and  the  generated  forces  will  be  identi- 
cal in  resisting  the  rotation  of  A,  produced  by  L.  All  of 
our  knowledge  of  mechanical  power  is  derived  from  exper- 
ience and  not  from  a  priori  reasoning,  and  we  must  follow 
the  same  teacher  when  we  substitute  the  moment  of  an 
"applied''  force  for  the  moment  of  a  "generated"  force.  A 
generated  tensile  force  and  the  generating  force  have  the 
same  sign,  and  if  both  were  applied  or  possessed  rotating 
power  they  would  rotate  the  beam  in  the  same  direction,  but 
a  generated  compressive  force  and  the  generating  force  have 
opposite  signs  and  tend  to  rotate  the  beam  in  opposite  di- 
rections. The  rule  for  the  substitution  then  is,  the  moment 
of  a  generated  tensile  force  is  the  moment  of  an  *  'applied' '  fotce 
in  the  opposite  direction,  and  the  moment  of  a  generated  com- 
pressive force  is  the  moment  of  an  "applied"  force  in  the 
same  direction.  The  "common  theory"  recognizes  the  cor- 
rectness of  this  rule  when  it  substitutes  the  moment  of  the 
"applied"  force  T,  Fig  i,  for  the  moment  of  the  opposite 
generated  tensile  force  in  A,  but  ignores  it,  and  thus  the 
teachings  of  experience  when  it  substitutes  the"  moment  of 
the  "applied"  force  C,  for  the  moment  of  the  opposite  gen- 
erated compressive  force  in  A,  and  thus  arrives  at  an  error- 
nious  conclusion.  For  resisting  the  rotation  of  A,  it  re- 
quires a  force  C  in  the  same  direction  with  T. 

Equilibrium  at  the  Section  of  Maximum  Stress. — Conceive 
the  beam  to  be  divided,  at  the  section  of  greatest  horizontal 
stress,  and  the  forces  to  be  balanced  in  each  part  as  claimed 
by  the  common  theory,  then  place  them  together  but  sepa- 
rated by  the  rubber  body  as  shown  in  Fig.  3.  The  lines  of 
action  of  the  (T  T)  and  (C  C)  forces  are  represented  as  not 
conciding  so  as  not  to  obscure  the  action  that  is  claimed  for 
them. 

Now  if  +  T  balances  —  C,  it  can  only  do  so  from  the 
body  A,  transmitting  the  auxiliary  shearing  force  of  -f  T, 
to  the  line  of  action  of  —  C,  and  this  becomes  +  C,  which 
is  the  shearing  of  -f  T,  and  if  —  T  balances  +  C,  it  can 
only  do  so  from  the  beam  B,  transmitting  the  shearing  force 


18 


ot  —  T,  to  the  line  of  action  of  +  C,  and  this  becomes  —  C. 
Then  the  two  forces  +  C  and  —  C  are  the  shearing  forces 
of  the  pair  of  T  forces  and  compress  the  rubber  body  j  ust 
as  if  they  had  been  originally  applied  to  it. 

The  common  theory  makes  —  T  sustain  one-half  of  the 
load  on  B  and  its  shearing  force  —  C,  one-half  of  that  on 
A.  Now  force  —  T,  and  its  shearing  —  C,  are  in  nature 
and  mechanics  the  same  identical  force,  this  theory  then 
makes  force  —  T  exhaust  it  power  to  sustain  the  load  twice 
over,  at  the  same  time,  or  in  other  words,  it  sustains  the 
whole  load  on  either  A  or  B,  which  is  true  in  this  method 


•r  +   T 


A  A 


Pis.  3 


of  balancing,  but  it  is  done  in  machines  directly,  without 
the  fiction  of  giving"the  shearing  forces  leverage. 

The  development  of  the  static  forces  (T  T)  must  be 
attended  by  their'equal  shearing  forces,  (C  C)  in  this  theory, 
or  they  can  possess  no  leverage  resistance  to  the  rotation  of 
the  loads  (L  L).  The  identity  of  the  pair  of  (T  T)  forces 
and  the  pair  of  (C  C)  in  this  equilibrium  follows  from  the 
' 'parallel ogran  of  forces";  for  produce  the  lines  of  action  of 
each  of  the  T  forces,  backward,  until  each  intersects  that 
of  its  load  I/.  From  these  points  draw  the  two  diagonals 
to  the  points  of  support  and  there  decompose  them  and  we 
will  obtain  the  pair  of  T  forces  which  now  become  a  pair 
C  forces.  This  also,  demonstrates  that  a  neutral  line  and 


an  uniformly  varying  compressive  force  cannot  exist  in  this 
condition  of  equilibrium. 

Resisting  Moment. — A  writer  states  that  at  any  section, 
"The  moment  of  the  external  force  or  forces  will  be  equal 
but  opposite  in  sign,  to  the  internal  resisting  moment."  A 
resistance  pure  and  simple  like  that  offered  by  a  rope  to 
extension  or  a  column  to  compression  can  have  no  resisting 
moment,  nothing  but  a  force  can  have  a  moment.  The 
internal  resistance  offered  by  the  material  of  the  beam  is 
the  passive  measure  of  the  intensity  of  the  forces.  The 
vertical  load  generates  the  horizontal  forces  through  its 
ability  to  rotate  the  beam,  the  material  passively  resists  this 
rotation  and  measures  the  intensify  of  the  horizontal  effort. 

No  error  would  result  in  calling  it  a  "resisting  moment" 
as  a  mark  of  identification,  but  when  it  is  further  said  that 
"the  sign  of  this  moment  must  be  opposite  to  that  of  the 
load"  that  produced  it,  a  fatal  error  is  introduced  from  an 
entire  misconception  of  the  subject;  for  it  is  conceded  that 
the  vertical  applied  forces  generate  the  horizontal  forces, 
each  on  its  own  side  of  the  section  of  maximum  stress. 

It  is  a  well  recognized  principle  that  like  produces  like 
and  not  unlike,  that  the  energy  of  any  movement  or  tend- 
ency is  balanced  by  that  which  it  generates  in  exhausting 
itself.  Then  to  require  the  load  on  A,  Fig.  i,  to  exhaust 
its  rotating  power  by  producing  a  right  handed  couple  (T 
C)  violates  undisputed  principles  of  nature. 

If  the  load  generated  nothing  but  rotation  and  the  balanc- 
ing "couple"  was  of  entirely  different  origin  the  statement 
would  be  true.  From  their  symmetrical  arrangement  the 
horizontal  forces,  at  the  section  of  maximum  stress,  has  the 
semblance  of  this  truth,  but  at  any  other  section  it  has  not 
even  this  semblance. 

Equilibrium  at  Any  Section. — All  writers  on  the  common 
theory  of  flexure  agree  that  at  any  section  "the  moment  of 
the  applied  or  deflecting  forces  equals  the  sum  of  the 
moments  of  the  resisting  forces. 

Then  let  us  consider  the  equilibrium,  at  the  quarter  sec- 


20 

tion  of  the  half  beam  A,  by  removing  the  left  hand  quarter 
of  the  beam  A  B  thus  giving  us  Fig.  4.  Between  this 
quarter  and  the  middle  section  the  molecular  movement  of 
the  material  is  to  the  left,  above  the  neutral  line,  and  to 
the  right  below  this  line.  The  direction  of  these  movements 
demonstrate  that  there  are  only  two  forces  acting  between 
these  sections,  one  pulling  the  material  of  the  beam  from 
and  the  other  pushing  it  to  the  middle  section.  The  forces 
t  and  c  acting  at  the  quarter  section  are  then  represented  in 
direction  by  the  arrows  in  Fig.  4,  and  there  are  no  others. 

In  the  "common  theory"  the  load  L  on  B  is  balanced  by 
the  left-handed  couple  (T  C)  at  the  middle  section,  and  all 
unbalanced  vertical  forces  being  thus  provided  for,  we  are 
lead  to  inquire  from  what  outside  source  do  the  forces  t  and  c 
come  and  what  do  they  represent.  They  are,  at  this  sec- 
tion the  generated  components  of  the  load  L  that  rested 
upon  the  removed  quarter  of  the  beam  A  B  and  represent 
it  in  the  system  of  forces  given  in  Fig.  4  and  the  moment 
of  the  removed  load,  though  forming  apparently  a  left- 
hande.d  couple  must  have  been  balanced,  in  the  whole  beam 
by  the  left-handed  couple  (/  c)  in  defiance  of  the  contrary 
law  of  statics — they  do  balance,  though  in  accordance  with 
our  rule  for  substituting  "applied"  for  generated  forces;  in 
order  to  determine  the  mechanical  moments  of  the  generated 
forces  /  and  c  when  resisting  the  moment  of  the  removed 
load  on  A,  the  center  of  rotation  being  at  the  bottom  of  the 
section 

In  discussing  this  equilibrium  the  "common  theory" 
writers  represent  a  right-handed  couple  as  acting  at  this 
section,  composed  of  forces  directly  opposite  in  direction  to 
/  and  c,  which  is  a  fiction,  as  we  have  j  List  shown  that  /  and 
c,  as  represented  in  Fig.  4,  are  the  only  horizontal  forces 
that  can  possibly  act  at  this  section  and  produce  the  '  'de- 
formation' '  that  is  known  to  take  place.  The  fiction  that 
there  exists  at  this  section  a  "resisting  moment  opposite  in 
sign"  composed  of  the  resistance  of  the  material  to  longi- 
tudinal tension  and  compression,  will  not  do,  for  this  resist- 


21 

ance  of  the  material  simply  measures  the  horizontal  intenA 
sity  of  /  and  <:,  and  being  passive,  from  the  very  definition 
can  have  no  moment  or  tendency  to  rotate  A.  One  class  of 
writers3  represent  a  right-handed  "strain  couple"  and  an- 
other4 a  right-handed  *  'force  couple"  as  acting  at  this  quar- 
ter section,  when  in  fact  it  is  left  handed  apparently. 

The  forces  /  and  c  cannot  balance  each  other,  and,  there- 
fore, cannot  be  generated  equal  at  this  section,  although  it 
is  a  fundamental  principle  of  the  common  theory  that  they 
should. 

We  have  seen  that  in  order  that  c  shall  be  balanced  by  t, 
the  beam  A  must  transfer  t  as  a  shearing  force  until  it  be- 
comes opposite  c  on  its  own  line  of  action;  this  it  cannot  do 


Fie.  4 


without  destroying  the  equality  that  is  claimed  to  exist  be- 
tween the  tension  and  compression  at  the  middle  section, 
for  with  a  shearing  force  t  acting  opposite  to  c,  at  the  quar- 
ter section,  the  resultant  compression  at  the  middle  section 
will  be  C — /  as  the  compressive  and  shearing  force  act  in 
opposite  directions  along  the  same  line  of  action, — the  re- 
sultant force  C — t  will  therefore  be  less  than  T. 

Then  as  no  shearing  force  can  act  in  the  opposite  direc- 
tion to  that  of  r,  at  this  or  any  other  section,  without  dim- 
inishing the  compression  on  the  middle  section,  these  tension 
and  compression  forces  cannot  balance  at  the  middle  or  sec- 
tion of  maximum  stress.  For  these  horizontal  T  and  C  forces 


(3)  Weisbach's  Mechanics  of  Eng.,  pp.  548  (Coxe). 

(4)  Wood's  Elements  of  Analytical  Mechanics,  pp.  129. 


22 


increase  from  zero  at  the  ends  to  their  maximum  value  at  the 
middle  section  of  the  beam.  When  the  lever- arm  of  the 
load  is  zero  the  horizontal  forces  are  zero,  then  each  incre- 
ment to  the  lever- arm  of  L,  gives  an  increment  to  the  value 
of  T  and  C  and  if  the  forces  cannot  balance  and  develope 
equally  at  the  section  of  their  origin,  the  opportunity  is  lost 
and  this  condition  cannot  be  imposed  upon  them  after  they 
come  into  existence,  and  this  they  cannot  do,  for  it  requires 
at  each  such  section  the  horizontal  effect  of  at  least  three 
forces  to  balance,  which  can  only  be  fulfilled  at  the  middle 
section. 

Spring  Experiments. — Certain  experiments  made  by 
cutting  the  beam  into  two  pieces  and  reconnecting  the  parts 
by  placing  elastic  springs  between  them  and  opposite  the 
resultants  of  the  tension  and  compression  of  Fig.  3,  thus 
giving  it  the  semblance  of  the  original,  and  the  develope- 
ment  of  equal  tension  and  compression  in  the  springs  has 
caused  some  minds  to  conclude  that  the  demonstration  of 
the  common  theory  "has  all  the  absoluteness  of  geometry.'' 

Prof.  Rankine  says  of  the  right  line  balance  of  parallel 
forces  when  applied  to  a  body,  "that  all  pairs  of  directly 
opposed  equal  forces  may  be  left  out  of  consideration,  for 
each  such  pair  is  independently  balanced  what  soever  its 
position  may  be."  The  horizontal  forces  in  the  whole  beam 
consists  of  two  such  directly  opposed  pairs,  and  the  body 
as  a  whole  must  remain  at  rest  whether  the  forces  of  these 
pairs  are  equal  or  not;  for  there  is  no  unbalanced  force  to 
cause  horizontal  motion.  Now,  if  the  whole  is  at  rest  with- 
out the  forces  of  these  paris  being  equal,  must  not  the  half 
of  the  whole  be  in  the  same  condition?  Then  "the  com- 
mon theory"  equality  of  the  tension  and  compression  in  the 
beam,  and  its  semblance,  when  its  parts  are  connected  by 
springs,  cannot  be  a  necessity  from  the  first  condition  of 
equilibrium,  but  results  from  an  entirely  different  mechan- 
ical principle. 

Moment  is  the  result  of  the  arrest  of  rotary  motion — 
when  we  pull  a  rope  with  a  force  at  each  end  right  line 


23 

motion  is  arrested  but  neither  force  possesses  any  leverage, 
but  should  these  equal  and  opposite  tensile  forces  be  devel- 
oped in  arresting  rotary  motion,  each  would  possess  lever- 
age in  resisting  the  cause  of  the  rotation  in  proportion  to 
their  distance  from  the  center.  This  leverage  it  can  only 
have  from  the  bodys  transmitting  a  shearing  force  to  the 
center  of  rotation. 

In  the  spring  experiments  the  tension  spring  arrests  the 
rotary  motion  and  each  half-beam  transmits  to  the  compres- 
sion springs,  the  centre  of  rotation,  equal  and  opposite 
shearing  forces  (in  order  to  possess  leverage)  which  causes 
the  compression  to  develope  equal  to  the  tension,  which  is 
not  a  result  of  the  first  condition. 

The  half  beams  A  and  B,  Fig.  3,  connected  by  springs 
possess  another  property  that  is  fatal  to  the  supposed  iden- 
tity between  it  and  the  solid  beam.  After  placing  a  pair  of 
springs  at  the  middle,  divide  the  half  beam  into  two  parts 
at  any  section  and  reconnect  them  by  means  of  another  pair 
of  springs  on  the  same  horizontal  line  with  the  first  pair, 
then  demonstration  and  experiment  shows  that  the  stress 
in  each  pair  is  the  same,  thus  making  the  tension  and  com- 
pression uniform,  from  end  to  end,  in  the  common  theory 
beam,  when  in  the  beam  as  it  is,  they  are  zero  [at  the  ends 
and  a  maximum  at  some  interior  section. 

This  uniformity  of  stress,  from  end  to  end  of  the  beam, 
in  the  *  'common  theory,"  results  from  one  of  the  oldest 
known  principles  of  statics,  "the  parallelogram  forces." 
In  Fig.  i,  prolong  the  lines  of  the  forces  Land  T  until  they 
intersect,  then  the  resultant  would  move  A  in  the  direction 
of  its  diagonal,  which  must  be  resisted  by  decomposing  the 
resultant  into  its  original  components  JL,  and  T,  and  apply 
C  and  S  equal  and  directly  opposite  and  equilibrium  will 
be  established.  If  we  decompose  the  resultant  of  L,  and  T 
at  any  point  on  the  diagonal  of  A,  we  will  obtain  the  orig- 
inal components;  that  is,  the  tension  T  does  not  vary  in 
value.  If  we  now  make  the  same  diagram  of  forces  foi  the 
right  hand  half  of  the  beam  B,  and  place  them  together  as 


24 

in  Fig.  3,  with  a  spring  opposite  to  the  line  of  action  of  the 
tensile  forces  and  remove  the  applied  T  forces,  then  by  ap- 
plying the  loads  Iy  to  both  A  and  B,  the  upper  ends  of  the 
diagonals,  being  now  connected,  they  will  generate  an  uni- 
form tensile  stress  between  these  points.  And  there  can  be 
no  equilibrium  of  moments  between  the  deflecting  and 
resisting  forces  at  any  point  in  the  beam,  except  at  the  sec- 
tion of  maximum  stress. 

This  failure  of  the  stresses  in  "the  common  theory" 
beam  to  decrease  longitudinally  from  the  section  of  maxi- 
mum stress,  results  from  another  "well  recognized"  princi- 
ple of  statics.  An  applied  force  distributes  itself  through 
the  body  on  its  line  of  action  without  increase  or  decrease, 
and  its  point  of  application  may  be  taken  anywhere  on  this 
line  in  the  body.  The  applied  forces  T  and  C,  Fig.  i, 
must  follow  this  law  and  be  of  uniform  tensity  from  end  to 
end  of  A. 

Here  again  it  is  seen  that  the  common  theory  adopts  a 
process  that  leads  to  results  that  are  entirely  the  reverse  of 
what  is  true  for  the  beam. 

Center  of  Rotation.  From  moments  measuring  the  tend- 
ency of  forces  to  produce  rotation  about  a  point,  there  must 
be,  in  every  body  that  is  acted  upon  by  a  system  of  forces 
that  balance  through  the  equality  of  their  moments,  a  com- 
mon center  of  such  tendency  to  rotation,  which  determines 
the  lengths  of  the  lever-arms  of  the  forces.  And  if  the 
equilibrium  of  this  system  is  thoroughly  understood,  the 
amount  and  character  of  the  work  performed  by  each  indi- 
vidual force  must  be  clearly  ascertained — to  know  the  tool 
that  it  works  with,  its  lever-arm  or  distance  from  the  cenier 
of  rotation  must  be  accurately  determined.  If  the  line  of 
direction  of  a  force  passes  through  the  center  of  rotation, 
the  lenth  of  its  lever-arm  is  zero,  then  by  giving  it  a  mo- 
ment about  some  other  point,  we  give  it  what  it  does  not 
possess.  This  is  the  condition  in  the  equilibrium  of  the 
common  theory,  the  horizontal  forces  develop  equal,  from 
the  pressure  producing  forces  passing  through  the  center  of 


25 

rotation.  When  this  is  the  case  '  'we  may  take  our  origin 
of  moments  anywhere, ' '  but  the  results  obtained  express  a 
mathematical  play  instead  of  a  mechanical  condition. 

An  algbraic  equation  may  express  the  equilibrium  as  an 
arithmetical  problem,  but  entirely  misrepresent  the  mechan- 
ical condition.  Had  the  common  theory  writers  determined 
the  actual  center  of  rotation  and  formed  their  equation  to 
express  the  mechanical  conditions  about  this  center,  instead 
of  as  the  R.  R.  Gazette*  says  as  an  apology  for  not  doing 
so,  the  arithmetical  results  for  the  couple  being  '  'constant 
wherever  be  the  origin  of  moments,"  which  "shows  that  it 
is  not  necessary  to  assign  the  origin  of  moments, ' '  it  would 
have  condemned  and  shown  the  absurdities  of  the  theory. 
To  obtain  this  constant  arithmetical  result,  when  the  origin 
is  between  the  lines  of  action  of  the  forces  of  the  couple, 
their  moments,  being  of  the  same  sign,  add;  but  when  the 
center  is  outside,  the  moments  of  the, forces  must  subtract, 
as  they  then  have  opposite  signs,  in  order  to  maintain  this 
"constant"  arithmetical  result.  It  does  appear  very  im- 
portant that  we  determine  by  locating  the  center  of  rotation, 
whether  the  pair  of  forces  work  together  or  work  against 
each  other. 

The  center  of  rotation  has  been  variously  located  by  dif- 
ferent writers.  Wiesbach  says  it  is  on  the  neutral  line  in 
the  section  of  maximum  stress.  Another  writer, 5  after  locat- 
ing it  at  the  center  of  A  figure  i,  says  it  may  be  "any- 
where ;"  for  the  couples  (L  S)  and  (T  C)  "tend  to  turn  the 
half  beam  around  its  center  of  mass.  Equilibrium  being 
established,  the  tendency  to  turn  about  any  point  within 
or  without  the  beam  is  zero,  and  the  origin  of  moment  may 
be  anywhere."  The  motion  itself  may  be  zero,  but  the  ten- 
dency can  never  be  zero  while  the  forces  possess  leverage. 
Therefore  our  writer  must  let  it  remain  at  the  center  of 
mass  of  A,  as  it  cannot  be  removed  by  bad  logic  from  where 
worse  mechanics  located  it. 

(5)  R.  R.  Gazette,  September  6th,  1889. 


26 

Another  author6  says  :  "The  usual  method  of  failure  of 
beams  is  by  cross-breaking,  this  is  caused  by  the  external 
forces  producing  rotation  around  some  point  in  the  section 
of  failure ' ' ;  but  fails  to  locate  this  very  important  point. 

In  a  purely  static  system  of  forces,  such  as  the  ' '  common 
theory,"  the  center  of  rotation  is  easily  ascertained,  for  the 
shearing  of  each  force  travels  directly  to  this  point,  and 
the  intersection  of  their  lines  of  travel  locates  it  unerringly. 

In  the  "spring  experiments,"  and  all  other  devices  by 
which  it  is  demonstrated  that  the  tension  and  compression 
develop  equal,  the  compressed  spring,  or  whatever  repre- 
sents it,  is,  and  must  be,  for  this  equality,  the  center  of  ro- 
tation or  tendency.  Then  in  the  solid  beam,  the  beam  must 
be  able  to  rotate  around  a  point  on  the  compressive  force, 
which  it  cannot  and  has  no  appearance  of  doing  during 
bending,  therefore  the .  compression  and  tension  cannot  be 
generated  equal  in  the  solid  beam. 

But  in  the  beam  as  it  is,  we  have  only  a  vertical  shear- 
ing force,  which  alone  is  not  sufficient  to  locate  the 
center,,  and  we  must  combine  with  this  the  knowledge  de- 
rived from  the  character  of  the  ' '  deformation ' '  of  the  beam 
during  bending.  The  vertical  shearing  forces  locates  the 
center  of  rotation  in  the  section  of  maximum  stress,  and  as 
this  is  also  the  only  plane  that  remains  "  fixed,"  relatively, 
to  the  whole,  during  the  process  of  bending,  the  center  of 
rotation  must  be  a  point  of  this  plane  as  it  is  a  * '  fixed 
point"  also. 

If  we  divide  a  beam,  that  is  supported  at  its  middle  and 
loaded  at  its  ends,  into  very  thin  horizontal  layers,  each 
distinguished  from  the  other  by  differences  in  color,  then 
during  the  process  of  bending  the  points  of  each  layer  will 
be  observed  to  have  a  downward  motion,  and  that  the 
points  of  no  layer  can  rotate  around  a  point,  as  a  center,  in 
a  plane  higher  than  its  own— the  points  of  the  bottom  layer 
can  rotate  around  a  point  in  its  own  plane,  but  not  around 


(6)  Merriman's  Mechanics  of  Materials,  pp.  42. 


27 

a  point  in  a  plane  higher  than  its  own  ;  then,  as  there  can 
only  be  one  such  point  for  the  whole  beam,  it  must  be  at 
the  intersection  of  the  plane  of  maximum  stress  with  the 
plane  of  the  concave  or  compressed  side  of  the  beam,  where 
Galileo  determined  it  to  be  nearly  two  and  a  half  centuries 
ago. 

Development  of  Equilibrium. — When  an  unbalanced  force, 
such  as  a  vertical  load,  is  applid  to  a  beam  or  body,  and 
equilibrium  is  established  either  by  the  development  or  the 
application  of  forces  acting  at  right  angles  to  the  direction 
of  the  applied,  unbalanced  force,  two  cases  may  be  dis- 
tinguished : 

1 .  The  resultant  moment  is  the  moment  of  a  single  force 
in  the  direction  of  each  axis,  or  the  compressive  forces  have 
no  moment,  as  their  lines  of  action  pass  through  the  center 
of  rotation  or  tendency. 

2.  The  resultant  moment  in  the  direction  of  the  axis  of 
the  applied  force  only,  is  the  moment  of  a  single  force,  and 
in  the  direction  of  the  other  it  is  the  sum  of  the  moments  of 
two  components,  or  the  compressive  component  has  leverage 
as  well  as  the  tension,  and  neither  have  any  shearing  force. 
These  distinctions  the  writer  has  never  seen  made  by  any 
text  book  of  mechanics.     When  motion  could  take  place 
in  the  body,  or  half  body,  as  a  whole,  were  it  not  for  the 
conditions  of  ' '  fixedness  ' '  imposed  by  connections  outside 
of  the  body  itself,  then  the  arrest  of  the  rotary  motion  of 
the  body,  as  a  whole,   produces  tension   in  the  "fixing" 
farthest  from  the  center  of  rotation  and  equilibrium  of  mo- 
ments is  established  from  both  the  applied  force  and  the 
developed  tension  transmitting  their  shearing   forces  to  the 
center  of  rotation. 

In  Fig.  i  we  have  the  load  I/,  the  reaction  of  the  support 
S,  and  equal  and  opposed  to  S  the  shearing  force  L,,  thus 
giving  us  the  vertical  effect  of  three  forces,  the  resultant 
moment  of  which  is  that  of  the  single  force  I,,  for  as  both 
the  opposed  forces  S  and  shearing  I,  pass  through  the  cen- 


28 

ter  of  rotation,  their  lever-arm  is  zero,  consequently  the  mo- 
ment of  the  three  is  that  of  L.  An  analysis  of  the  equili- 
brium of  the  forces  acting  in  the  direction  of  the  other  axis 
or  the  (T  C)  forces  leads  to  the  same  conclusion,  for  if  they 
balance  they  do  not  dc  so  in  any  mysterious  mythical  way 
This  principle  is  the  basis  of  the  equilibrium  in  the  spring 
experiments,  nut-crackers,  the  "common  theory"  and  a 
host  of  other  contrivances  with  which  we  are  familiar. 

But  when  the  condition  of  "fixednes"  is  such  as  exists 
between  the  two  halves  of  a  beam,  and  each  half  is  solicited 
to  rotate  in  opposite  directions  around  a  fixed  center,  no 
motion  can  take  place  except  through  the  "deformation" 
of  the  beam  itself.  This  requires  that  its  rigedness  shall 
first  be  overcome  by  longitudinal  compression  in  order 
that  the  half-beam  may  assume  the  arc  form  first  by  the 
shorting  of  the  material  on  the  concave  side,  the  result  of 
which  is  bending  and  the  develoyment  of  tension.  The 
pressure  components  are  first  generated  and  are  not  the 
shearings  of  the  tensile  forces, — they  do  not  pass  through 
the  center  of  rotation  and  therefore  possess  leverage. 

With  equilibrium  established  for  any  given  load  L, 
under  our  first  case,  the  order  of  development  is  as  follows: 
An  increment  to  the  load  L,  gives  an  increment  to  the  rota- 
tion, its  conversion  into  a  tendency  gives  an  increment  to 
the  tension,  this  in  turn  gives  an  increment  to  its  shearing 
and  this  an  increment  to  the  moment  of  the  tensile  force  by 
which  equilibrium  is  re-established. 

In  the  second  case  an  increment  to  the  load  I,  gives  an 
increment  to  the  compression  which  gives  an  increment  to 
the  shortening,  this  admits  of  an  increment  to  the  bending 
(which  is  actual  rotation  and  not  a  tendency},  and  this 
finally  gives  an  increment  to  the  tension,  and  equilibrium 
is  re-established  by  the  increments  to  the  horizontal  forces 
giving  an  increment  to  the  sum  of  their  moments  without 
the  increments  balancing  as  right  line  forces. 


29 

THE  WRITER'S  THEORY. 

The  method  of  generating  the  horizontal  forces,  distin- 
guished in  our  second  case,  has  been  made  the  basis  of  the 
theory  given  in  my  book  on  "The  Strength  of  Beams  and 
Columns"7  The  Railroad  Gazettee  of  July  26,  1889,  in  its 
review  of  this  book  says:  "If  a  theory  containing  a  me- 
chanical absurdity  leads  to  rational  results,  it  does  not 
prove  that  the  "fallacy"  is  truth,  but  rather  that  some 
other  error  must  be  "involved  to  offset  the  first  error." 
The  Gazette  then  proceeds  to  demonstrate  the  existence  of 
the  fallacy  from  glittering  generalties  and  "mathematical 
necessities"  drawn  from  the  formulated  principles  of  pure 
statics  that  are  applicable,  solely  and  alone,  to  the  first  case 
above  given,  with  equal  propriety,  its  fallacy  might  have 
been  demonstrated  from  its  failure  to  comply  with  "the 
well  recognized  principles"  of  Dynamics,  Hydrostatics  or 
those  of  any  other  kindred  science.  This  method  is  cer- 
tainly not  within  the  scope  of  fair  criticism. 

The  horizontal  forces  do  not  exist  of  themselves,  neither 
do  we  apply  them;  they  are  generated.  An  unloaded  beam 
has  none,  and  if  the  load  be  applied  pound  by  pound,  the 
horizontal  forces  grow  and  strengthen  from  zero,  with  the 
increase  in  the  load,  and  the  method  of  their  generation  is 
the  vital  part  of  any  theory  of  \hzflexure  of  beams  and  col- 
umns. 

If  the  method  of  generation,  described  in  my  theory,  is 
impossible  mechanics,  it  can  be  demonstrated,  per  se,  and 
not  because  there  is  another  rational  method  that  is  appli- 
cable to  entirely  different  mechanical  conditions,  and  the 
conclusions  drawn  from  the  results  of  one  Iprocess  are  at 
variance  with  those  from  the  other. 

No  "common  theory"  writer,  with  whom  I  am  familiar, 
makes  the  process  of  generating  the  horizontal  forces  any 
part  of  his  demonstration.  The  conclusions  are  "mathe- 

(7)  E.  &  F.  N.  Spon,  publishers,  New  York  and  London. 


30 

matical  necessities,"  drawn  from  general  algebraic  equa- 
tions expressing  possible  equilibrium  about  every  point  in 
the  universe,  and  the  physical  condition  of  the  beam  is 
only  considered  when  the  amount  of  the  stresses  are  needed 
to  give  value  to  the  forces  of  the  general  equation. 

The  Gazette  again  says.  "Many  results  found  by  the 
author  agree  fairly  with  those  of  direct  experiment,  and  in 
some  cases  it  may  give  better  results  than  those  found  by 
the  common  theory."  Exact  identity  between  the  compu- 
ted strength,  from  my  theor}',  and  the  determined  experi- 
mental strength  could  not  be  expected  from  the  record  of 
experiments  already  made,  which  was  the  only  source  I 
had  from  which  to  draw  my  illustrations.  The  tensile  and 
compressive  strength,  as  recorded,  is  usually  the  mean  of 
many  tests,  the  single  test  varying  in  value  from  five  to  ten 
per  cent,  from  the  mean  values,  while  the  computed  tran- 
severse  strength  is  for  a  single  test,  which  test  strength  may 
vary  from  the  strength  computed  from  mean  values  five  to 
ten  per  cent.  Had  the  number  of  tests  been  the  same  to 
determine  the  tensile,  compressive  and  transverse  strength 
then  the  computed  strength,  from  the  mean  of  the  tensile 
and  compressive  tests,  would  be  identical  with  the  mean  of 
the  transverse  tests.  In  all  cases  where  the  strength  of  the 
beam  was  computed  from  the  tested  tensile  and  compressive 
strength  of  the  identical  material  contained  in  the  beam 
broken,  the  discrepency  was  three  per  cent,  and  less,  this  is 
especially  noticeable  in  the  very  careful  experiments  of  Mr. 
Kirkaldy  and  those  made  at  the  Watertown  Arsenal. 

Columns. — The  amount  of  the  load  applied  to  a  column 
directly  measures  the  amount  of  the  internal  stress,  and 
their  relation  is  the  crucial  test  of  the  correctness  of  any 
theory  of  flexure.  The  "common  theory"  has  utterly 
failed  to  establish  any  such  relation, — Gordon's,  the  most 
successful  formula,  is  based  upon  a  direct  denial  of  the  cor- 
rectness of  the  theory;  in  its  demonstration,  it  is  said,  8 

(8)  Burr's  Materials  of  Engineering,  pp.  430. 


31 

'The  condition  of  stress  in  any  normal  section  of  a  long 
column  is  that  of  a  uniformly  varying  system  composed  of 
a  uniform  stress  and  a  stress  couple, ' '  the  resultant  com- 
pression of  this  '  'system' '  is  then  greater  than  the  resultant 
tension,  which  moves  the  neutral  line  from  the  center  of 
gravity  of  the  section.  The  writers' s  theory  fully  estab- 
lishes this  relation,  the  correctness  of  which  is  amply  sus- 
tained by  all  recorded  experiments. 

R.  H.  COUSINS. 
Austin,  Texas,  August  29,  1891. 


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